We claim that \(\mathcal {J}\)  is measure-zero in \(\operatorname{Gr}(n,r)\) . For each \(\mathbf {P}\in \mathcal {C},\)  at least one eigenvalue of \(\operatorname{Hess}f(\mathbf {P})\)  is greater than 0. By hess Dphi, we know at least one eigenvalue of \(\text{D}\phi (\mathbf {P})\)  is greater than 1. By the first result of cs, the dimension of \(W_\mathbf {P}\)  is strictly less than the dimension of \(\operatorname{Gr}(n,r)\) .
So \(W_\mathbf {P}\)  is measure-zero in \(\operatorname{Gr}(n,r)\)  by [1]}.
Given that the step size satisfies \(0 < \gamma \le \Gamma ^{**}\) , the map \(\phi \)  is a local diffeomorphism by local diff, and so is \(\phi ^t\)  for all \(t \ge 0\) . Thus their pre-images of measure-zero sets have zero measure.
As \((\phi ^{t})^{-1} \equiv \phi ^{-t}\) , the set \(\phi ^{-t} (W_\mathbf {P})\)  has zero measure for each \(t \ge 0\) .
Then \(\mathcal {J}\) 
is a countable union of measure-zero sets, so has zero measure.
